I found mine when I bought my Fender American Std Jazz, Ampeg SVT3 Pro head and added an Sansamp DI pedal. Simple setup that gets compliments whenever I play out. The Sansamp was the icing. FWIW i use Ernie Ball Power Slinky Roundwound Bass Strings 55-110.

I'm totally green to the idea of having "a sound", but I would hazard to say that it's a lifelong journey.

When you start out, you hear the big changes and you make big changes to achieve what you hear. As your ears mature, you begin to notice the finer nuances and you keep tweaking with less and less obvious changes to the sound.

I expect the sounds you like and are looking for would change as you mature too -

Personally, my journey to the sound I like has consisted of playing a cheap Ibanez RD300 stock through a Roland 30W cube.
I changed the amp to a fender rumble 350w 2x10" to get the extra power I needed, and struggled with boominess (my fault as it turns out)
I then changed my pickups to Nordstrands in a Neck J and Bridge MM, and I really began to dig the Jazz tone.
I bought a sadowsky outboard preamp, and dug the J sound it gave me even more.
I then started listening to more funk, and realised I could get a very adequate burp tone out of the MM by changing my playing position to over the bridge pup.
I then upgraded my amp to a TCE RH450 + RS210, and am in the process of working out the tone that I like from this with/without the sadowsky, with/without tubetone/spectracomp effects.
I'm now waiting on my first set of flatwounds, T-Is and La Bellas to investigate further.

In fact, having written all of that, "a sound" is probably just as much an evolution based on gear you go through in your journey as it is a maturing of your ear.

I've been playing for 30 years and have been fighting my sound since day one. I've tried almost every bass you can imagine, most major brands of amps and cabinets, various pickups and configurations, but it wasn't until I discovered the Tech 21 line of products, specifically the Bass Driver DI, that I really became satisfied. I love that company!

I used to have a Fender bassman 100 and 4x12s so I was on my way.. Then one night at practice my amp blew up and all I had sitting around were a Yamaha GQ2031 31 band graphic eq and an old QSC power amp. The eq I set for -10db and the output +4db so I used the eq as my boost and the amp pushed it. With 31 bands you could dial in your sound, which I did. I used that for ages with everyone laughing saying I couldn't use that for a bass rig. Then when they heard it they shut up.

Finally another friend who I had been jamming with decided to sell off his late 70s SVT to get a Hartke 3500. I had a lot of the money down and my friend's sound man needed some bass bins I had so I told him to pay off my friend for what I owed on the amp - which everyone agreed to. Then I lucked into an Ampeg V9 (9x10) cabinet and that was my sound for years....

When I stopped looking, it found me. We've been inseparable ever since.

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There is something to be said for this. I watched and listened to all kinds of video clips. Bought the bass, cabinets, and strings I liked. Went to GC and played some cabs and amps. Bought one amp. Was somewhat satisfied, even though the strings were not broken in. Then I saw a video by Ed Friedland and that sold me on the amp on which I decided to commit.

At that point (with the new amp in process of shipping) I was feeling more than a bit unsure if I would ever find "The Tone In My Head" (TTIMH).

After reading many similar threads here on TB, I decided that whatever resulted from the combination I was building, THAT would be My Sound. It may not be TTIMH, but it would be a very good, if not excellent, sound.

Besides, the gear that gets you TTIYH in your living room or practice space under one set of conditions, may not deliver TTIYH in various other venues or conditions, or at higher (or lower) volumes.

Once I made that decision, I became much more relaxed about gear and such. Now, the strings (flats) are really mellowing and, when I think about it, I can almost hear TTIMH in my living room and in my group's jam space. (But I try to ignore it.)

I am not currently gigging, but whatever tone I get when I do, I am sure I'll be happy with it.

Somehow my tone doesn't really change much from amp to amp, only bass to bass. Sure amps may make it sound quieter, louder, muddier, boomey, ect, but the underlying tone is still there. My tone changes greatly from bass to bass though and I enjoy playing a lot of different basses often. My favorite though has been my precious Fender Jazz with Bartolini pickups that I chanced upon on CL, though a Fender Rumble 100 amp.

I don't have a sound, it just sound like a fretted bass or a fretless bass and the EQ change every day so.

Why ? well, there is two answers :

1) I discovered one time that I had a wonderful tone in one jam and the next day without changing a thing it sounded like s*** ... so I started to realize the best thing to do is to have a workable sound and not trying to have a sig tone like Marcus Miller.

2) I started to play multipart harmony with one bass and to do so I needed each bass part to sound different enough that you could tell appart or the blend is musical. So again my "unique" tone is just gone and to this day I think the tone isn't important at all, what you play is the important part.

Just over years and a bit of gear.
I had nice amps, good basses etc, but still wasn't what I really wanted. I even have Tech 21 gear and didn't like that sound either.
Eventually I got an amp that I really love and then I ended up getting another bass. This one was more boutique and pricey and I really liked the sound. So I was fully in love with my sound, but I was still missing the grit I needed if I wanted a good distortion. I then read on hear about Darkglass B7K, I got one, and hey presto, my sound.

It takes a long time, and unfortunately for some people (like me), a lot of money (but it doesn't have to at all).

Just try lots of gear, and try them together, the bass you think has the sound, through the amp you think you want. I could save a lot of "not quite" moments